The trend component suggests a decreasing trend,
with an average value of -0.2 and a standard deviation of 0.28 .
The seasonal component suggests a repeating pattern,
with an average value of 0 and a standard deviation of 0.05 .
The remainder component suggests random fluctuations,
with an average value of 0 and a standard deviation of 0.28 .
The time series appears to be additive.
Augmented Dickey-Fuller Test Results:
Test Statistic: -8.271031
P-value: 0.01
Critical Values:
The time series is stationary based on the ADF test.
Augmented Dickey-Fuller Test Results After Differencing:
Test Statistic: -30.68056
P-value: 0.01
Critical Values:
The time series is stationary based on the ADF test.
The trend component suggests a decreasing trend,
with an average value of -0.03 and a standard deviation of 0.28 .
The seasonal component suggests a repeating pattern,
with an average value of 0 and a standard deviation of 0.04 .
The remainder component suggests random fluctuations,
with an average value of 0 and a standard deviation of 0.29 .
The time series appears to be additive.
Augmented Dickey-Fuller Test Results:
Test Statistic: -7.594511
P-value: 0.01
Critical Values:
The time series is stationary based on the ADF test.
Augmented Dickey-Fuller Test Results After Differencing:
Test Statistic: -30.42139
P-value: 0.01
Critical Values:
The time series is stationary based on the ADF test.
The trend component suggests a increasing trend,
with an average value of 0.13 and a standard deviation of 0.35 .
The seasonal component suggests a repeating pattern,
with an average value of 0 and a standard deviation of 0.05 .
The remainder component suggests random fluctuations,
with an average value of 0.01 and a standard deviation of 0.32 .
The time series appears to be additive.
Augmented Dickey-Fuller Test Results:
Test Statistic: -6.248758
P-value: 0.01
Critical Values:
The time series is stationary based on the ADF test.
Augmented Dickey-Fuller Test Results After Differencing:
Test Statistic: -29.35235
P-value: 0.01
Critical Values:
The time series is stationary based on the ADF test.
The trend component suggests a increasing trend,
with an average value of 5.91 and a standard deviation of 2.95 .
The seasonal component suggests a repeating pattern,
with an average value of 0 and a standard deviation of 0.04 .
The remainder component suggests random fluctuations,
with an average value of 0 and a standard deviation of 0.34 .
The time series appears to be additive.
Augmented Dickey-Fuller Test Results:
Test Statistic: -2.358925
P-value: 0.4262639
Critical Values:
The time series is not stationary based on the ADF test.
Augmented Dickey-Fuller Test Results After Differencing:
Test Statistic: -22.84948
P-value: 0.01
Critical Values:
The time series is stationary based on the ADF test.
The trend component suggests a increasing trend,
with an average value of 5.94 and a standard deviation of 2.87 .
The seasonal component suggests a repeating pattern,
with an average value of 0 and a standard deviation of 0.04 .
The remainder component suggests random fluctuations,
with an average value of 0 and a standard deviation of 0.31 .
The time series appears to be additive.
Augmented Dickey-Fuller Test Results:
Test Statistic: -2.016143
P-value: 0.5715748
Critical Values:
The time series is not stationary based on the ADF test.
Augmented Dickey-Fuller Test Results After Differencing:
Test Statistic: -22.66937
P-value: 0.01
Critical Values:
The time series is stationary based on the ADF test.
The trend component suggests a increasing trend,
with an average value of 6.26 and a standard deviation of 3.02 .
The seasonal component suggests a repeating pattern,
with an average value of 0 and a standard deviation of 0.05 .
The remainder component suggests random fluctuations,
with an average value of 0 and a standard deviation of 0.32 .
The time series appears to be additive.
Augmented Dickey-Fuller Test Results:
Test Statistic: -3.1987
P-value: 0.08796191
Critical Values:
The time series is not stationary based on the ADF test.
Augmented Dickey-Fuller Test Results After Differencing:
Test Statistic: -20.60433
P-value: 0.01
Critical Values:
The time series is stationary based on the ADF test.
The trend component suggests a increasing trend,
with an average value of 140.28 and a standard deviation of 77.1 .
The seasonal component suggests a repeating pattern,
with an average value of 0 and a standard deviation of 0.14 .
The remainder component suggests random fluctuations,
with an average value of -0.01 and a standard deviation of 0.67 .
The time series appears to be additive.
Augmented Dickey-Fuller Test Results:
Test Statistic: -1.625921
P-value: 0.736648
Critical Values:
The time series is not stationary based on the ADF test.
Augmented Dickey-Fuller Test Results After Differencing:
Test Statistic: -5.055112
P-value: 0.01
Critical Values:
The time series is stationary based on the ADF test.
The trend component suggests a increasing trend,
with an average value of 11515.29 and a standard deviation of 5265.6 .
The seasonal component suggests a repeating pattern,
with an average value of -0.41 and a standard deviation of 41.98 .
The remainder component suggests random fluctuations,
with an average value of -0.16 and a standard deviation of 120.32 .
The time series appears to be additive.
Augmented Dickey-Fuller Test Results:
Test Statistic: -1.773162
P-value: 0.670995
Critical Values:
The time series is not stationary based on the ADF test.
Augmented Dickey-Fuller Test Results After Differencing:
Test Statistic: -7.392571
P-value: 0.01
Critical Values:
The time series is stationary based on the ADF test.
The trend component suggests a increasing trend,
with an average value of 116.8 and a standard deviation of 2.12 .
The seasonal component suggests a repeating pattern,
with an average value of 0.08 and a standard deviation of 1.4 .
The remainder component suggests random fluctuations,
with an average value of 0.29 and a standard deviation of 2.27 .
The time series appears to be additive.
Augmented Dickey-Fuller Test Results:
Test Statistic: -1.982863
P-value: 0.5855674
Critical Values:
The time series is not stationary based on the ADF test.
Augmented Dickey-Fuller Test Results After Differencing:
Test Statistic: -10.6259
P-value: 0.01
Critical Values:
The time series is stationary based on the ADF test.
The trend component suggests a increasing trend,
with an average value of 4.88 and a standard deviation of 3.61 .
The seasonal component suggests a repeating pattern,
with an average value of 0 and a standard deviation of 0.07 .
The remainder component suggests random fluctuations,
with an average value of -0.01 and a standard deviation of 0.77 .
The time series appears to be additive.
Augmented Dickey-Fuller Test Results:
Test Statistic: -2.777681
P-value: 0.2491291
Critical Values:
The time series is not stationary based on the ADF test.
Augmented Dickey-Fuller Test Results After Differencing:
Test Statistic: -6.888716
P-value: 0.01
Critical Values:
The time series is stationary based on the ADF test.
The trend component suggests a increasing trend,
with an average value of 7.75 and a standard deviation of 3.25 .
The seasonal component suggests a repeating pattern,
with an average value of 0 and a standard deviation of 0.06 .
The remainder component suggests random fluctuations,
with an average value of 0 and a standard deviation of 0.33 .
The time series appears to be additive.
Augmented Dickey-Fuller Test Results:
Test Statistic: -2.183748
P-value: 0.5005302
Critical Values:
The time series is not stationary based on the ADF test.
Augmented Dickey-Fuller Test Results After Differencing:
Test Statistic: -14.02196
P-value: 0.01
Critical Values:
The time series is stationary based on the ADF test.
The trend component suggests a increasing trend,
with an average value of 5.98 and a standard deviation of 1.59 .
The seasonal component suggests a repeating pattern,
with an average value of 0 and a standard deviation of 0.1 .
The remainder component suggests random fluctuations,
with an average value of 0 and a standard deviation of 0.56 .
The time series appears to be additive.
Augmented Dickey-Fuller Test Results:
Test Statistic: -3.162679
P-value: 0.09436521
Critical Values:
The time series is not stationary based on the ADF test.
Augmented Dickey-Fuller Test Results After Differencing:
Test Statistic: -8.081753
P-value: 0.01
Critical Values:
The time series is stationary based on the ADF test.
The chart shows the Consumer Price Index (CPI) and its moving averages (MA10, MA50, MA100) all trending upwards, indicating long-term inflation growth. The MA10 closely follows the CPI’s short-term fluctuations, while the MA50 and MA100 smooth out these variations, lagging behind and highlighting the broader trend. The recent steep climb in the CPI suggests an acceleration in inflation not yet fully captured by the moving averages.
The chart illustrates the Gross Domestic Product (GDP) and its associated moving averages (MA10, MA50, MA100), all ascending, which signifies sustained economic growth. The MA10 is tracking the GDP’s immediate shifts, reflecting the economy’s short-term variations, whereas the MA50 and MA100 provide a more subdued perspective, trailing behind and emphasizing the long-term economic trend. The noticeable upward trajectory in the most recent GDP data points to a potential surge in economic activity, a detail that the longer-term moving averages have yet to fully incorporate.
The chart shows the US Dollar Index with its moving averages (MA10, MA50, MA100), reflecting a volatile currency market. Short-term fluctuations are closely followed by MA10, while MA50 and MA100 smooth out these fluctuations to reveal more sustained trends. A notable peak and subsequent decline around 2020 are followed by a recovery, with recent trends suggesting an increase in the dollar’s value.
The graph shows the trend of 30-year mortgage rates from around 1980 to 2020, with a clear peak in the early 1980s. It includes the actual rates and three moving averages (50-day, 100-day, and 500-day), which smooth out short-term fluctuations. Over time, there’s a notable decline in mortgage rates, with all the lines converging, indicating stabilization of rates towards 2020.
The graph displays the historical trends of the unemployment rate from the 1960s to around 2020, including actual rates and their moving averages over 10, 50, and 100 periods. The chart shows several spikes indicating periods of increased unemployment, with the most significant peaks likely corresponding to economic recessions. Notably, there is an extreme spike in unemployment at the end of the timeline, suggesting a sudden, sharp increase in unemployment, possibly caused by the Coronavirus pandamic.
Source Code
---title: "Exploratory Data Analysis"subtitle: "Time Series Plot, Lag Plot, ACF, PACF, and Augmented Dickey-Fuller Test"format: html: smooth-scroll: true toc: true code-fold: true code-tools: true embed-resources: true mermaid: theme: neutralbibliography: citation.bibexecute: echo: false warning: false---```{r setup, include=FALSE}knitr::opts_chunk$set(warning=FALSE, message=FALSE)library(zoo)library(TTR)library(readr)library(readxl)library(ggplot2)library(forecast)library(tseries)library(xts)library(tidyverse)```## Treasury Security::: {.panel-tabset}## 3 Month Treasury Bill```{r fig.width=8, fig.height=4}df <-read_excel('data/data/sr_t-bill-rates.xlsx')df$observation_date <-as.Date(df$observation_date)columns <-c("T3MFF")for (col in columns) { xts_data <-xts(df[, col], order.by = df$observation_date) ts_data <-ts(df[, col], start=c(1982, 1, 4), frequency=252) decomposed <-decompose(ts_data)plot(decomposed$x, main =paste("Original Time Series:", col), xlab ='Date', ylab ='Value')plot(decomposed$trend, main ="Trend Component", xlab ='Date', ylab ='Value')plot(decomposed$seasonal, main ="Seasonal Component", xlab ='Date', ylab ='Value',ylim =c(-0.5,0.5))plot(decomposed$random, main ="Remainder Component", xlab ='Date', ylab ='Value')lag.plot(ts_data, lags=1, do.lines=FALSE, main="Lag Plot") acf_data <-acf(ts_data, main=paste("ACF Plot:", col)) dacf_data <-pacf(ts_data, main=paste("PACF Plot:", col)) diff_ts_data <-diff(ts_data)acf(diff_ts_data, main=paste("ACF After Differencing:", col))pacf(diff_ts_data, main=paste("PACF After Differencing:", col)) }for (col in columns) { trend <- decomposed$trendif (!is.null(trend)) { trend_mean <-mean(trend, na.rm =TRUE) trend_sd <-sd(trend, na.rm =TRUE) trend_trend <-ifelse(trend_mean >0, "increasing", "decreasing")cat("The trend component suggests a", trend_trend, "trend, \nwith an average value of", round(trend_mean, 2), "and a standard deviation of", round(trend_sd, 2), ".\n\n") } else {cat("There is no discernible trend component.\n\n") } seasonal <- decomposed$seasonalif (!is.null(seasonal)) { seasonal_mean <-mean(seasonal, na.rm =TRUE) seasonal_sd <-sd(seasonal, na.rm =TRUE)cat("The seasonal component suggests a repeating pattern, \nwith an average value of", round(seasonal_mean, 2), "and a standard deviation of", round(seasonal_sd, 2), ".\n\n") } else {cat("There is no discernible seasonal component.\n\n") } remainder <- decomposed$randomif (!is.null(remainder)) { remainder_mean <-mean(remainder, na.rm =TRUE) remainder_sd <-sd(remainder, na.rm =TRUE)cat("The remainder component suggests random fluctuations, \nwith an average value of", round(remainder_mean, 2), "and a standard deviation of", round(remainder_sd, 2), ".\n\n") } else {cat("There is no discernible remainder component.\n\n") }if (decomposed$type =="multiplicative") {cat("The time series appears to be multiplicative.\n\n") } else {cat("The time series appears to be additive.\n\n") } adf_test <-adf.test(ts_data)cat("Augmented Dickey-Fuller Test Results:\n")cat("Test Statistic:", adf_test$statistic, "\n")cat("P-value:", adf_test$p.value, "\n")cat("Critical Values:", adf_test$critical, "\n")if (adf_test$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n")} adf_test_diff <-adf.test(diff_ts_data)cat("Augmented Dickey-Fuller Test Results After Differencing:\n")cat("Test Statistic:", adf_test_diff$statistic, "\n")cat("P-value:", adf_test_diff$p.value, "\n")cat("Critical Values:", adf_test_diff$critical, "\n")if (adf_test_diff$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n") } }```## 6 Month Treasury Bill```{r fig.width=8, fig.height=4}df <-read_excel('data/data/sr_t-bill-rates.xlsx')df$observation_date <-as.Date(df$observation_date)columns <-c("T6MFF")for (col in columns) { xts_data <-xts(df[, col], order.by = df$observation_date) ts_data <-ts(df[, col], start=c(1982, 1, 4), frequency=252) decomposed <-decompose(ts_data)plot(decomposed$x, main =paste("Original Time Series:", col), xlab ='Date', ylab ='Value')plot(decomposed$trend, main ="Trend Component", xlab ='Date', ylab ='Value')plot(decomposed$seasonal, main ="Seasonal Component", xlab ='Date', ylab ='Value',ylim =c(-0.5,0.5))plot(decomposed$random, main ="Remainder Component", xlab ='Date', ylab ='Value')lag.plot(ts_data, lags=1, do.lines=FALSE, main="Lag Plot") acf_data <-acf(ts_data, main=paste("ACF Plot:", col)) dacf_data <-pacf(ts_data, main=paste("PACF Plot:", col)) diff_ts_data <-diff(ts_data)acf(diff_ts_data, main=paste("ACF After Differencing:", col))pacf(diff_ts_data, main=paste("PACF After Differencing:", col)) }for (col in columns) { trend <- decomposed$trendif (!is.null(trend)) { trend_mean <-mean(trend, na.rm =TRUE) trend_sd <-sd(trend, na.rm =TRUE) trend_trend <-ifelse(trend_mean >0, "increasing", "decreasing")cat("The trend component suggests a", trend_trend, "trend, \nwith an average value of", round(trend_mean, 2), "and a standard deviation of", round(trend_sd, 2), ".\n\n") } else {cat("There is no discernible trend component.\n\n") } seasonal <- decomposed$seasonalif (!is.null(seasonal)) { seasonal_mean <-mean(seasonal, na.rm =TRUE) seasonal_sd <-sd(seasonal, na.rm =TRUE)cat("The seasonal component suggests a repeating pattern, \nwith an average value of", round(seasonal_mean, 2), "and a standard deviation of", round(seasonal_sd, 2), ".\n\n") } else {cat("There is no discernible seasonal component.\n\n") } remainder <- decomposed$randomif (!is.null(remainder)) { remainder_mean <-mean(remainder, na.rm =TRUE) remainder_sd <-sd(remainder, na.rm =TRUE)cat("The remainder component suggests random fluctuations, \nwith an average value of", round(remainder_mean, 2), "and a standard deviation of", round(remainder_sd, 2), ".\n\n") } else {cat("There is no discernible remainder component.\n\n") }if (decomposed$type =="multiplicative") {cat("The time series appears to be multiplicative.\n\n") } else {cat("The time series appears to be additive.\n\n") } adf_test <-adf.test(ts_data)cat("Augmented Dickey-Fuller Test Results:\n")cat("Test Statistic:", adf_test$statistic, "\n")cat("P-value:", adf_test$p.value, "\n")cat("Critical Values:", adf_test$critical, "\n")if (adf_test$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n")} adf_test_diff <-adf.test(diff_ts_data)cat("Augmented Dickey-Fuller Test Results After Differencing:\n")cat("Test Statistic:", adf_test_diff$statistic, "\n")cat("P-value:", adf_test_diff$p.value, "\n")cat("Critical Values:", adf_test_diff$critical, "\n")if (adf_test_diff$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n") } }```## 1 Year Treasury Bill```{r fig.width=8, fig.height=4}df <-read_excel('data/data/sr_t-bill-rates.xlsx')df$observation_date <-as.Date(df$observation_date)columns <-c("T1YFF")for (col in columns) { xts_data <-xts(df[, col], order.by = df$observation_date) ts_data <-ts(df[, col], start=c(1982, 1, 4), frequency=252) decomposed <-decompose(ts_data)plot(decomposed$x, main =paste("Original Time Series:", col), xlab ='Date', ylab ='Value')plot(decomposed$trend, main ="Trend Component", xlab ='Date', ylab ='Value')plot(decomposed$seasonal, main ="Seasonal Component", xlab ='Date', ylab ='Value',ylim =c(-0.5,0.5))plot(decomposed$random, main ="Remainder Component", xlab ='Date', ylab ='Value')lag.plot(ts_data, lags=1, do.lines=FALSE, main="Lag Plot") acf_data <-acf(ts_data, main=paste("ACF Plot:", col)) dacf_data <-pacf(ts_data, main=paste("PACF Plot:", col)) diff_ts_data <-diff(ts_data)acf(diff_ts_data, main=paste("ACF After Differencing:", col))pacf(diff_ts_data, main=paste("PACF After Differencing:", col)) }for (col in columns) { trend <- decomposed$trendif (!is.null(trend)) { trend_mean <-mean(trend, na.rm =TRUE) trend_sd <-sd(trend, na.rm =TRUE) trend_trend <-ifelse(trend_mean >0, "increasing", "decreasing")cat("The trend component suggests a", trend_trend, "trend, \nwith an average value of", round(trend_mean, 2), "and a standard deviation of", round(trend_sd, 2), ".\n\n") } else {cat("There is no discernible trend component.\n\n") } seasonal <- decomposed$seasonalif (!is.null(seasonal)) { seasonal_mean <-mean(seasonal, na.rm =TRUE) seasonal_sd <-sd(seasonal, na.rm =TRUE)cat("The seasonal component suggests a repeating pattern, \nwith an average value of", round(seasonal_mean, 2), "and a standard deviation of", round(seasonal_sd, 2), ".\n\n") } else {cat("There is no discernible seasonal component.\n\n") } remainder <- decomposed$randomif (!is.null(remainder)) { remainder_mean <-mean(remainder, na.rm =TRUE) remainder_sd <-sd(remainder, na.rm =TRUE)cat("The remainder component suggests random fluctuations, \nwith an average value of", round(remainder_mean, 2), "and a standard deviation of", round(remainder_sd, 2), ".\n\n") } else {cat("There is no discernible remainder component.\n\n") }if (decomposed$type =="multiplicative") {cat("The time series appears to be multiplicative.\n\n") } else {cat("The time series appears to be additive.\n\n") } adf_test <-adf.test(ts_data)cat("Augmented Dickey-Fuller Test Results:\n")cat("Test Statistic:", adf_test$statistic, "\n")cat("P-value:", adf_test$p.value, "\n")cat("Critical Values:", adf_test$critical, "\n")if (adf_test$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n")} adf_test_diff <-adf.test(diff_ts_data)cat("Augmented Dickey-Fuller Test Results After Differencing:\n")cat("Test Statistic:", adf_test_diff$statistic, "\n")cat("P-value:", adf_test_diff$p.value, "\n")cat("Critical Values:", adf_test_diff$critical, "\n")if (adf_test_diff$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n") } }``````{r}df <-read_excel('data/data/t-bill-vs-econ-index.xlsx',col_types =c("date","numeric","numeric","numeric","numeric","numeric","numeric","numeric","numeric","numeric"))df$observation_date <-as.Date(df$observation_date)selected_columns <-c("DGS10", "DGS20", "DGS30", "MORTGAGE30", "CPI", "FEDFUNDS", "USDIDX", "GDP", "UER")df <- df[!apply(is.na(df[selected_columns]) | df[selected_columns] =="", 1, all), ]```## 10 Year Treasury Bond```{r fig.width=8, fig.height=4}columns <-c("DGS10")df_10 <- df[!is.na(df[columns]) & df[columns] !="", ]for (col in columns) { xts_data <-xts(df_10[, col], order.by = df_10$observation_date) ts_data <-ts(df_10[, col], start=c(1962, 1, 2), frequency=252) decomposed <-decompose(ts_data)plot(decomposed$x, main =paste("Original Time Series:", col), xlab ='Date', ylab ='Value')plot(decomposed$trend, main ="Trend Component", xlab ='Date', ylab ='Value')plot(decomposed$seasonal, main ="Seasonal Component", xlab ='Date', ylab ='Value')plot(decomposed$random, main ="Remainder Component", xlab ='Date', ylab ='Value')lag.plot(ts_data, lags=1, do.lines=FALSE, main="Lag Plot") acf_data <-acf(ts_data, main=paste("ACF Plot:", col)) dacf_data <-pacf(ts_data, main=paste("PACF Plot:", col)) diff_ts_data <-diff(ts_data)acf(diff_ts_data, main=paste("ACF After Differencing:", col))pacf(diff_ts_data, main=paste("PACF After Differencing:", col)) }for (col in columns) { trend <- decomposed$trendif (!is.null(trend)) { trend_mean <-mean(trend, na.rm =TRUE) trend_sd <-sd(trend, na.rm =TRUE) trend_trend <-ifelse(trend_mean >0, "increasing", "decreasing")cat("The trend component suggests a", trend_trend, "trend, \nwith an average value of", round(trend_mean, 2), "and a standard deviation of", round(trend_sd, 2), ".\n\n") } else {cat("There is no discernible trend component.\n\n") } seasonal <- decomposed$seasonalif (!is.null(seasonal)) { seasonal_mean <-mean(seasonal, na.rm =TRUE) seasonal_sd <-sd(seasonal, na.rm =TRUE)cat("The seasonal component suggests a repeating pattern, \nwith an average value of", round(seasonal_mean, 2), "and a standard deviation of", round(seasonal_sd, 2), ".\n\n") } else {cat("There is no discernible seasonal component.\n\n") } remainder <- decomposed$randomif (!is.null(remainder)) { remainder_mean <-mean(remainder, na.rm =TRUE) remainder_sd <-sd(remainder, na.rm =TRUE)cat("The remainder component suggests random fluctuations, \nwith an average value of", round(remainder_mean, 2), "and a standard deviation of", round(remainder_sd, 2), ".\n\n") } else {cat("There is no discernible remainder component.\n\n") }if (decomposed$type =="multiplicative") {cat("The time series appears to be multiplicative.\n\n") } else {cat("The time series appears to be additive.\n\n") } adf_test <-adf.test(ts_data)cat("Augmented Dickey-Fuller Test Results:\n")cat("Test Statistic:", adf_test$statistic, "\n")cat("P-value:", adf_test$p.value, "\n")cat("Critical Values:", adf_test$critical, "\n")if (adf_test$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n")} adf_test_diff <-adf.test(diff_ts_data)cat("Augmented Dickey-Fuller Test Results After Differencing:\n")cat("Test Statistic:", adf_test_diff$statistic, "\n")cat("P-value:", adf_test_diff$p.value, "\n")cat("Critical Values:", adf_test_diff$critical, "\n")if (adf_test_diff$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n") } }```## 20 Year Treasury Bond```{r fig.width=8, fig.height=4}columns <-c("DGS20")df_20 <- df[!is.na(df[columns]) & df[columns] !="", ]for (col in columns) { xts_data <-xts(df_20[, col], order.by = df_20$observation_date) ts_data <-ts(df_20[, col], start=c(1962, 1, 2), frequency=252) decomposed <-decompose(ts_data)plot(decomposed$x, main =paste("Original Time Series:", col), xlab ='Date', ylab ='Value')plot(decomposed$trend, main ="Trend Component", xlab ='Date', ylab ='Value')plot(decomposed$seasonal, main ="Seasonal Component", xlab ='Date', ylab ='Value')plot(decomposed$random, main ="Remainder Component", xlab ='Date', ylab ='Value')lag.plot(ts_data, lags=1, do.lines=FALSE, main="Lag Plot") acf_data <-acf(ts_data, main=paste("ACF Plot:", col)) dacf_data <-pacf(ts_data, main=paste("PACF Plot:", col)) diff_ts_data <-diff(ts_data)acf(diff_ts_data, main=paste("ACF After Differencing:", col))pacf(diff_ts_data, main=paste("PACF After Differencing:", col)) }for (col in columns) { trend <- decomposed$trendif (!is.null(trend)) { trend_mean <-mean(trend, na.rm =TRUE) trend_sd <-sd(trend, na.rm =TRUE) trend_trend <-ifelse(trend_mean >0, "increasing", "decreasing")cat("The trend component suggests a", trend_trend, "trend, \nwith an average value of", round(trend_mean, 2), "and a standard deviation of", round(trend_sd, 2), ".\n\n") } else {cat("There is no discernible trend component.\n\n") } seasonal <- decomposed$seasonalif (!is.null(seasonal)) { seasonal_mean <-mean(seasonal, na.rm =TRUE) seasonal_sd <-sd(seasonal, na.rm =TRUE)cat("The seasonal component suggests a repeating pattern, \nwith an average value of", round(seasonal_mean, 2), "and a standard deviation of", round(seasonal_sd, 2), ".\n\n") } else {cat("There is no discernible seasonal component.\n\n") } remainder <- decomposed$randomif (!is.null(remainder)) { remainder_mean <-mean(remainder, na.rm =TRUE) remainder_sd <-sd(remainder, na.rm =TRUE)cat("The remainder component suggests random fluctuations, \nwith an average value of", round(remainder_mean, 2), "and a standard deviation of", round(remainder_sd, 2), ".\n\n") } else {cat("There is no discernible remainder component.\n\n") }if (decomposed$type =="multiplicative") {cat("The time series appears to be multiplicative.\n\n") } else {cat("The time series appears to be additive.\n\n") } adf_test <-adf.test(ts_data)cat("Augmented Dickey-Fuller Test Results:\n")cat("Test Statistic:", adf_test$statistic, "\n")cat("P-value:", adf_test$p.value, "\n")cat("Critical Values:", adf_test$critical, "\n")if (adf_test$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n")} adf_test_diff <-adf.test(diff_ts_data)cat("Augmented Dickey-Fuller Test Results After Differencing:\n")cat("Test Statistic:", adf_test_diff$statistic, "\n")cat("P-value:", adf_test_diff$p.value, "\n")cat("Critical Values:", adf_test_diff$critical, "\n")if (adf_test_diff$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n") } }```## 30 Year Treasury Bond```{r fig.width=8, fig.height=4}columns <-c("DGS30")df_30 <- df[!is.na(df[columns]) & df[columns] !="", ]for (col in columns) { xts_data <-xts(df_30[, col], order.by = df_30$observation_date) ts_data <-ts(df_30[, col], start=c(1977, 2, 15), frequency=252) decomposed <-decompose(ts_data)plot(decomposed$x, main =paste("Original Time Series:", col), xlab ='Date', ylab ='Value')plot(decomposed$trend, main ="Trend Component", xlab ='Date', ylab ='Value')plot(decomposed$seasonal, main ="Seasonal Component", xlab ='Date', ylab ='Value')plot(decomposed$random, main ="Remainder Component", xlab ='Date', ylab ='Value')lag.plot(ts_data, lags=1, do.lines=FALSE, main="Lag Plot") acf_data <-acf(ts_data, main=paste("ACF Plot:", col)) dacf_data <-pacf(ts_data, main=paste("PACF Plot:", col)) diff_ts_data <-diff(ts_data)acf(diff_ts_data, main=paste("ACF After Differencing:", col))pacf(diff_ts_data, main=paste("PACF After Differencing:", col)) }for (col in columns) { trend <- decomposed$trendif (!is.null(trend)) { trend_mean <-mean(trend, na.rm =TRUE) trend_sd <-sd(trend, na.rm =TRUE) trend_trend <-ifelse(trend_mean >0, "increasing", "decreasing")cat("The trend component suggests a", trend_trend, "trend, \nwith an average value of", round(trend_mean, 2), "and a standard deviation of", round(trend_sd, 2), ".\n\n") } else {cat("There is no discernible trend component.\n\n") } seasonal <- decomposed$seasonalif (!is.null(seasonal)) { seasonal_mean <-mean(seasonal, na.rm =TRUE) seasonal_sd <-sd(seasonal, na.rm =TRUE)cat("The seasonal component suggests a repeating pattern, \nwith an average value of", round(seasonal_mean, 2), "and a standard deviation of", round(seasonal_sd, 2), ".\n\n") } else {cat("There is no discernible seasonal component.\n\n") } remainder <- decomposed$randomif (!is.null(remainder)) { remainder_mean <-mean(remainder, na.rm =TRUE) remainder_sd <-sd(remainder, na.rm =TRUE)cat("The remainder component suggests random fluctuations, \nwith an average value of", round(remainder_mean, 2), "and a standard deviation of", round(remainder_sd, 2), ".\n\n") } else {cat("There is no discernible remainder component.\n\n") }if (decomposed$type =="multiplicative") {cat("The time series appears to be multiplicative.\n\n") } else {cat("The time series appears to be additive.\n\n") } adf_test <-adf.test(ts_data)cat("Augmented Dickey-Fuller Test Results:\n")cat("Test Statistic:", adf_test$statistic, "\n")cat("P-value:", adf_test$p.value, "\n")cat("Critical Values:", adf_test$critical, "\n")if (adf_test$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n")} adf_test_diff <-adf.test(diff_ts_data)cat("Augmented Dickey-Fuller Test Results After Differencing:\n")cat("Test Statistic:", adf_test_diff$statistic, "\n")cat("P-value:", adf_test_diff$p.value, "\n")cat("Critical Values:", adf_test_diff$critical, "\n")if (adf_test_diff$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n") } }```:::## Economic Indicator::: {.panel-tabset}## CPI```{r fig.width=8, fig.height=4}columns <-c("CPI")df_cpi <- df[!is.na(df[columns]) & df[columns] !="", ]for (col in columns) { xts_data <-xts(df_cpi[, col], order.by = df_cpi$observation_date) ts_data <-ts(df_cpi[, col], start=c(1962, 2, 1), frequency=12) decomposed <-decompose(ts_data)plot(decomposed$x, main =paste("Original Time Series:", col), xlab ='Date', ylab ='Value')plot(decomposed$trend, main ="Trend Component", xlab ='Date', ylab ='Value')plot(decomposed$seasonal, main ="Seasonal Component", xlab ='Date', ylab ='Value',ylim =c(-0.5,0.5))plot(decomposed$random, main ="Remainder Component", xlab ='Date', ylab ='Value')lag.plot(ts_data, lags=1, do.lines=FALSE, main="Lag Plot") acf_data <-acf(ts_data, main=paste("ACF Plot:", col)) dacf_data <-pacf(ts_data, main=paste("PACF Plot:", col)) diff_ts_data <-diff(ts_data)acf(diff_ts_data, main=paste("ACF After Differencing:", col))pacf(diff_ts_data, main=paste("PACF After Differencing:", col)) }for (col in columns) { trend <- decomposed$trendif (!is.null(trend)) { trend_mean <-mean(trend, na.rm =TRUE) trend_sd <-sd(trend, na.rm =TRUE) trend_trend <-ifelse(trend_mean >0, "increasing", "decreasing")cat("The trend component suggests a", trend_trend, "trend, \nwith an average value of", round(trend_mean, 2), "and a standard deviation of", round(trend_sd, 2), ".\n\n") } else {cat("There is no discernible trend component.\n\n") } seasonal <- decomposed$seasonalif (!is.null(seasonal)) { seasonal_mean <-mean(seasonal, na.rm =TRUE) seasonal_sd <-sd(seasonal, na.rm =TRUE)cat("The seasonal component suggests a repeating pattern, \nwith an average value of", round(seasonal_mean, 2), "and a standard deviation of", round(seasonal_sd, 2), ".\n\n") } else {cat("There is no discernible seasonal component.\n\n") } remainder <- decomposed$randomif (!is.null(remainder)) { remainder_mean <-mean(remainder, na.rm =TRUE) remainder_sd <-sd(remainder, na.rm =TRUE)cat("The remainder component suggests random fluctuations, \nwith an average value of", round(remainder_mean, 2), "and a standard deviation of", round(remainder_sd, 2), ".\n\n") } else {cat("There is no discernible remainder component.\n\n") }if (decomposed$type =="multiplicative") {cat("The time series appears to be multiplicative.\n\n") } else {cat("The time series appears to be additive.\n\n") } adf_test <-adf.test(ts_data)cat("Augmented Dickey-Fuller Test Results:\n")cat("Test Statistic:", adf_test$statistic, "\n")cat("P-value:", adf_test$p.value, "\n")cat("Critical Values:", adf_test$critical, "\n")if (adf_test$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n")} adf_test_diff <-adf.test(diff_ts_data)cat("Augmented Dickey-Fuller Test Results After Differencing:\n")cat("Test Statistic:", adf_test_diff$statistic, "\n")cat("P-value:", adf_test_diff$p.value, "\n")cat("Critical Values:", adf_test_diff$critical, "\n")if (adf_test_diff$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n") } }```## GDP```{r fig.width=8, fig.height=4}columns <-c("GDP")df_gdp <- df[!is.na(df[columns]) & df[columns] !="", ]for (col in columns) { xts_data <-xts(df_gdp[, col], order.by = df_gdp$observation_date) ts_data <-ts(df_gdp[, col], start=c(1962, 10, 1), frequency=4) decomposed <-decompose(ts_data)plot(decomposed$x, main =paste("Original Time Series:", col), xlab ='Date', ylab ='Value')plot(decomposed$trend, main ="Trend Component", xlab ='Date', ylab ='Value')plot(decomposed$seasonal, main ="Seasonal Component", xlab ='Date', ylab ='Value')plot(decomposed$random, main ="Remainder Component", xlab ='Date', ylab ='Value')lag.plot(ts_data, lags=1, do.lines=FALSE, main="Lag Plot") acf_data <-acf(ts_data, main=paste("ACF Plot:", col)) dacf_data <-pacf(ts_data, main=paste("PACF Plot:", col)) diff_ts_data <-diff(ts_data)acf(diff_ts_data, main=paste("ACF After Differencing:", col))pacf(diff_ts_data, main=paste("PACF After Differencing:", col)) }for (col in columns) { trend <- decomposed$trendif (!is.null(trend)) { trend_mean <-mean(trend, na.rm =TRUE) trend_sd <-sd(trend, na.rm =TRUE) trend_trend <-ifelse(trend_mean >0, "increasing", "decreasing")cat("The trend component suggests a", trend_trend, "trend, \nwith an average value of", round(trend_mean, 2), "and a standard deviation of", round(trend_sd, 2), ".\n\n") } else {cat("There is no discernible trend component.\n\n") } seasonal <- decomposed$seasonalif (!is.null(seasonal)) { seasonal_mean <-mean(seasonal, na.rm =TRUE) seasonal_sd <-sd(seasonal, na.rm =TRUE)cat("The seasonal component suggests a repeating pattern, \nwith an average value of", round(seasonal_mean, 2), "and a standard deviation of", round(seasonal_sd, 2), ".\n\n") } else {cat("There is no discernible seasonal component.\n\n") } remainder <- decomposed$randomif (!is.null(remainder)) { remainder_mean <-mean(remainder, na.rm =TRUE) remainder_sd <-sd(remainder, na.rm =TRUE)cat("The remainder component suggests random fluctuations, \nwith an average value of", round(remainder_mean, 2), "and a standard deviation of", round(remainder_sd, 2), ".\n\n") } else {cat("There is no discernible remainder component.\n\n") }if (decomposed$type =="multiplicative") {cat("The time series appears to be multiplicative.\n\n") } else {cat("The time series appears to be additive.\n\n") } adf_test <-adf.test(ts_data)cat("Augmented Dickey-Fuller Test Results:\n")cat("Test Statistic:", adf_test$statistic, "\n")cat("P-value:", adf_test$p.value, "\n")cat("Critical Values:", adf_test$critical, "\n")if (adf_test$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n")} adf_test_diff <-adf.test(diff_ts_data)cat("Augmented Dickey-Fuller Test Results After Differencing:\n")cat("Test Statistic:", adf_test_diff$statistic, "\n")cat("P-value:", adf_test_diff$p.value, "\n")cat("Critical Values:", adf_test_diff$critical, "\n")if (adf_test_diff$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n") } }```## USD Index```{r fig.width=8, fig.height=4}columns <-c("USDIDX")df_usd <- df[!is.na(df[columns]) & df[columns] !="", ]for (col in columns) { xts_data <-xts(df_usd[, col], order.by = df_usd$observation_date) ts_data <-ts(df_usd[, col], start=c(2018, 9, 17), frequency=365) decomposed <-decompose(ts_data)plot(decomposed$x, main =paste("Original Time Series:", col), xlab ='Date', ylab ='Value')plot(decomposed$trend, main ="Trend Component", xlab ='Date', ylab ='Value')plot(decomposed$seasonal, main ="Seasonal Component", xlab ='Date', ylab ='Value')plot(decomposed$random, main ="Remainder Component", xlab ='Date', ylab ='Value')lag.plot(ts_data, lags=1, do.lines=FALSE, main="Lag Plot") acf_data <-acf(ts_data, main=paste("ACF Plot:", col)) dacf_data <-pacf(ts_data, main=paste("PACF Plot:", col)) diff_ts_data <-diff(ts_data)acf(diff_ts_data, main=paste("ACF After Differencing:", col))pacf(diff_ts_data, main=paste("PACF After Differencing:", col)) }for (col in columns) { trend <- decomposed$trendif (!is.null(trend)) { trend_mean <-mean(trend, na.rm =TRUE) trend_sd <-sd(trend, na.rm =TRUE) trend_trend <-ifelse(trend_mean >0, "increasing", "decreasing")cat("The trend component suggests a", trend_trend, "trend, \nwith an average value of", round(trend_mean, 2), "and a standard deviation of", round(trend_sd, 2), ".\n\n") } else {cat("There is no discernible trend component.\n\n") } seasonal <- decomposed$seasonalif (!is.null(seasonal)) { seasonal_mean <-mean(seasonal, na.rm =TRUE) seasonal_sd <-sd(seasonal, na.rm =TRUE)cat("The seasonal component suggests a repeating pattern, \nwith an average value of", round(seasonal_mean, 2), "and a standard deviation of", round(seasonal_sd, 2), ".\n\n") } else {cat("There is no discernible seasonal component.\n\n") } remainder <- decomposed$randomif (!is.null(remainder)) { remainder_mean <-mean(remainder, na.rm =TRUE) remainder_sd <-sd(remainder, na.rm =TRUE)cat("The remainder component suggests random fluctuations, \nwith an average value of", round(remainder_mean, 2), "and a standard deviation of", round(remainder_sd, 2), ".\n\n") } else {cat("There is no discernible remainder component.\n\n") }if (decomposed$type =="multiplicative") {cat("The time series appears to be multiplicative.\n\n") } else {cat("The time series appears to be additive.\n\n") } adf_test <-adf.test(ts_data)cat("Augmented Dickey-Fuller Test Results:\n")cat("Test Statistic:", adf_test$statistic, "\n")cat("P-value:", adf_test$p.value, "\n")cat("Critical Values:", adf_test$critical, "\n")if (adf_test$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n")} adf_test_diff <-adf.test(diff_ts_data)cat("Augmented Dickey-Fuller Test Results After Differencing:\n")cat("Test Statistic:", adf_test_diff$statistic, "\n")cat("P-value:", adf_test_diff$p.value, "\n")cat("Critical Values:", adf_test_diff$critical, "\n")if (adf_test_diff$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n") } }```## Fed Funds Rate```{r fig.width=8, fig.height=4}columns <-c("FEDFUNDS")df_ffr <- df[!is.na(df[columns]) & df[columns] !="", ]for (col in columns) { xts_data <-xts(df_ffr[, col], order.by = df_ffr$observation_date) ts_data <-ts(df_ffr[, col], start=c(1962, 2, 1), frequency=12) decomposed <-decompose(ts_data)plot(decomposed$x, main =paste("Original Time Series:", col), xlab ='Date', ylab ='Value')plot(decomposed$trend, main ="Trend Component", xlab ='Date', ylab ='Value')plot(decomposed$seasonal, main ="Seasonal Component", xlab ='Date', ylab ='Value')plot(decomposed$random, main ="Remainder Component", xlab ='Date', ylab ='Value')lag.plot(ts_data, lags=1, do.lines=FALSE, main="Lag Plot") acf_data <-acf(ts_data, main=paste("ACF Plot:", col)) dacf_data <-pacf(ts_data, main=paste("PACF Plot:", col)) diff_ts_data <-diff(ts_data)acf(diff_ts_data, main=paste("ACF After Differencing:", col))pacf(diff_ts_data, main=paste("PACF After Differencing:", col)) }for (col in columns) { trend <- decomposed$trendif (!is.null(trend)) { trend_mean <-mean(trend, na.rm =TRUE) trend_sd <-sd(trend, na.rm =TRUE) trend_trend <-ifelse(trend_mean >0, "increasing", "decreasing")cat("The trend component suggests a", trend_trend, "trend, \nwith an average value of", round(trend_mean, 2), "and a standard deviation of", round(trend_sd, 2), ".\n\n") } else {cat("There is no discernible trend component.\n\n") } seasonal <- decomposed$seasonalif (!is.null(seasonal)) { seasonal_mean <-mean(seasonal, na.rm =TRUE) seasonal_sd <-sd(seasonal, na.rm =TRUE)cat("The seasonal component suggests a repeating pattern, \nwith an average value of", round(seasonal_mean, 2), "and a standard deviation of", round(seasonal_sd, 2), ".\n\n") } else {cat("There is no discernible seasonal component.\n\n") } remainder <- decomposed$randomif (!is.null(remainder)) { remainder_mean <-mean(remainder, na.rm =TRUE) remainder_sd <-sd(remainder, na.rm =TRUE)cat("The remainder component suggests random fluctuations, \nwith an average value of", round(remainder_mean, 2), "and a standard deviation of", round(remainder_sd, 2), ".\n\n") } else {cat("There is no discernible remainder component.\n\n") }if (decomposed$type =="multiplicative") {cat("The time series appears to be multiplicative.\n\n") } else {cat("The time series appears to be additive.\n\n") } adf_test <-adf.test(ts_data)cat("Augmented Dickey-Fuller Test Results:\n")cat("Test Statistic:", adf_test$statistic, "\n")cat("P-value:", adf_test$p.value, "\n")cat("Critical Values:", adf_test$critical, "\n")if (adf_test$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n")} adf_test_diff <-adf.test(diff_ts_data)cat("Augmented Dickey-Fuller Test Results After Differencing:\n")cat("Test Statistic:", adf_test_diff$statistic, "\n")cat("P-value:", adf_test_diff$p.value, "\n")cat("Critical Values:", adf_test_diff$critical, "\n")if (adf_test_diff$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n") } }```## 30-Year Mortgage Rate```{r fig.width=8, fig.height=4}columns <-c("MORTGAGE30")df_mort <- df[!is.na(df[columns]) & df[columns] !="", ]for (col in columns) { xts_data <-xts(df_mort[, col], order.by = df_mort$observation_date) ts_data <-ts(df_mort[, col], start=c(1971, 4, 2), frequency=52) decomposed <-decompose(ts_data)plot(decomposed$x, main =paste("Original Time Series:", col), xlab ='Date', ylab ='Value')plot(decomposed$trend, main ="Trend Component", xlab ='Date', ylab ='Value')plot(decomposed$seasonal, main ="Seasonal Component", xlab ='Date', ylab ='Value')plot(decomposed$random, main ="Remainder Component", xlab ='Date', ylab ='Value')lag.plot(ts_data, lags=1, do.lines=FALSE, main="Lag Plot") acf_data <-acf(ts_data, main=paste("ACF Plot:", col)) dacf_data <-pacf(ts_data, main=paste("PACF Plot:", col)) diff_ts_data <-diff(ts_data)acf(diff_ts_data, main=paste("ACF After Differencing:", col))pacf(diff_ts_data, main=paste("PACF After Differencing:", col)) }for (col in columns) { trend <- decomposed$trendif (!is.null(trend)) { trend_mean <-mean(trend, na.rm =TRUE) trend_sd <-sd(trend, na.rm =TRUE) trend_trend <-ifelse(trend_mean >0, "increasing", "decreasing")cat("The trend component suggests a", trend_trend, "trend, \nwith an average value of", round(trend_mean, 2), "and a standard deviation of", round(trend_sd, 2), ".\n\n") } else {cat("There is no discernible trend component.\n\n") } seasonal <- decomposed$seasonalif (!is.null(seasonal)) { seasonal_mean <-mean(seasonal, na.rm =TRUE) seasonal_sd <-sd(seasonal, na.rm =TRUE)cat("The seasonal component suggests a repeating pattern, \nwith an average value of", round(seasonal_mean, 2), "and a standard deviation of", round(seasonal_sd, 2), ".\n\n") } else {cat("There is no discernible seasonal component.\n\n") } remainder <- decomposed$randomif (!is.null(remainder)) { remainder_mean <-mean(remainder, na.rm =TRUE) remainder_sd <-sd(remainder, na.rm =TRUE)cat("The remainder component suggests random fluctuations, \nwith an average value of", round(remainder_mean, 2), "and a standard deviation of", round(remainder_sd, 2), ".\n\n") } else {cat("There is no discernible remainder component.\n\n") }if (decomposed$type =="multiplicative") {cat("The time series appears to be multiplicative.\n\n") } else {cat("The time series appears to be additive.\n\n") } adf_test <-adf.test(ts_data)cat("Augmented Dickey-Fuller Test Results:\n")cat("Test Statistic:", adf_test$statistic, "\n")cat("P-value:", adf_test$p.value, "\n")cat("Critical Values:", adf_test$critical, "\n")if (adf_test$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n")} adf_test_diff <-adf.test(diff_ts_data)cat("Augmented Dickey-Fuller Test Results After Differencing:\n")cat("Test Statistic:", adf_test_diff$statistic, "\n")cat("P-value:", adf_test_diff$p.value, "\n")cat("Critical Values:", adf_test_diff$critical, "\n")if (adf_test_diff$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n") } }```## Unemployment Rate```{r fig.width=8, fig.height=4}columns <-c("UER")df_uemp <- df[!is.na(df[columns]) & df[columns] !="", ]for (col in columns) { xts_data <-xts(df_uemp[, col], order.by = df_uemp$observation_date) ts_data <-ts(df_uemp[, col], start=c(1962, 2, 1), frequency=12) decomposed <-decompose(ts_data)plot(decomposed$x, main =paste("Original Time Series:", col), xlab ='Date', ylab ='Value')plot(decomposed$trend, main ="Trend Component", xlab ='Date', ylab ='Value')plot(decomposed$seasonal, main ="Seasonal Component", xlab ='Date', ylab ='Value')plot(decomposed$random, main ="Remainder Component", xlab ='Date', ylab ='Value')lag.plot(ts_data, lags=1, do.lines=FALSE, main="Lag Plot") acf_data <-acf(ts_data, main=paste("ACF Plot:", col)) dacf_data <-pacf(ts_data, main=paste("PACF Plot:", col)) diff_ts_data <-diff(ts_data)acf(diff_ts_data, main=paste("ACF After Differencing:", col))pacf(diff_ts_data, main=paste("PACF After Differencing:", col)) }for (col in columns) { trend <- decomposed$trendif (!is.null(trend)) { trend_mean <-mean(trend, na.rm =TRUE) trend_sd <-sd(trend, na.rm =TRUE) trend_trend <-ifelse(trend_mean >0, "increasing", "decreasing")cat("The trend component suggests a", trend_trend, "trend, \nwith an average value of", round(trend_mean, 2), "and a standard deviation of", round(trend_sd, 2), ".\n\n") } else {cat("There is no discernible trend component.\n\n") } seasonal <- decomposed$seasonalif (!is.null(seasonal)) { seasonal_mean <-mean(seasonal, na.rm =TRUE) seasonal_sd <-sd(seasonal, na.rm =TRUE)cat("The seasonal component suggests a repeating pattern, \nwith an average value of", round(seasonal_mean, 2), "and a standard deviation of", round(seasonal_sd, 2), ".\n\n") } else {cat("There is no discernible seasonal component.\n\n") } remainder <- decomposed$randomif (!is.null(remainder)) { remainder_mean <-mean(remainder, na.rm =TRUE) remainder_sd <-sd(remainder, na.rm =TRUE)cat("The remainder component suggests random fluctuations, \nwith an average value of", round(remainder_mean, 2), "and a standard deviation of", round(remainder_sd, 2), ".\n\n") } else {cat("There is no discernible remainder component.\n\n") }if (decomposed$type =="multiplicative") {cat("The time series appears to be multiplicative.\n\n") } else {cat("The time series appears to be additive.\n\n") } adf_test <-adf.test(ts_data)cat("Augmented Dickey-Fuller Test Results:\n")cat("Test Statistic:", adf_test$statistic, "\n")cat("P-value:", adf_test$p.value, "\n")cat("Critical Values:", adf_test$critical, "\n")if (adf_test$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n")} adf_test_diff <-adf.test(diff_ts_data)cat("Augmented Dickey-Fuller Test Results After Differencing:\n")cat("Test Statistic:", adf_test_diff$statistic, "\n")cat("P-value:", adf_test_diff$p.value, "\n")cat("Critical Values:", adf_test_diff$critical, "\n")if (adf_test_diff$p.value <0.05) {cat("The time series is stationary based on the ADF test.\n\n") } else {cat("The time series is not stationary based on the ADF test.\n\n") } }```:::## Moving Average Smoothing::: {.panel-tabset}## CPIThe chart shows the Consumer Price Index (CPI) and its moving averages (MA10, MA50, MA100) all trending upwards, indicating long-term inflation growth. The MA10 closely follows the CPI's short-term fluctuations, while the MA50 and MA100 smooth out these variations, lagging behind and highlighting the broader trend. The recent steep climb in the CPI suggests an acceleration in inflation not yet fully captured by the moving averages.```{r}df <-read_excel('data/data/cpi-u.xlsx')df <-na.omit(df)df$Date <-as.Date(df[[1]])df$CPI <-as.numeric(df[[2]])df$MA10 <-rollmean(df$CPI, 10 , fill =NA, align ='right')df$MA50 <-rollmean(df$CPI, 50, fill =NA, align ='right')df$MA100 <-rollmean(df$CPI, 100, fill =NA, align ='right')plot(df$Date, df$CPI, main='CPI Moving Average', ylab='CPI', xlab='Year', type='l')lines(df$Date, df$MA10, col='blue')lines(df$Date, df$MA50, col='red')lines(df$Date, df$MA100, col='green')legend("bottomright", # Position of the legendlegend =c("CPI", "MA10", "MA50", "MA100"), # The text in the legendcol =c("black", "blue", "red", "green"), # The colorslty =1, # The line typescex =0.8) # The size of the legend textcat()```## GDPThe chart illustrates the Gross Domestic Product (GDP) and its associated moving averages (MA10, MA50, MA100), all ascending, which signifies sustained economic growth. The MA10 is tracking the GDP's immediate shifts, reflecting the economy's short-term variations, whereas the MA50 and MA100 provide a more subdued perspective, trailing behind and emphasizing the long-term economic trend. The noticeable upward trajectory in the most recent GDP data points to a potential surge in economic activity, a detail that the longer-term moving averages have yet to fully incorporate.```{r}df <-read_excel('data/data/GDP.xlsx')df <-na.omit(df)df$Date <-as.Date(df[[1]])df$GDP <-as.numeric(df[[2]])df$MA10 <-rollmean(df$GDP, 10, fill =NA, align ='right')df$MA50 <-rollmean(df$GDP, 50, fill =NA, align ='right')df$MA100 <-rollmean(df$GDP, 100, fill =NA, align ='right')plot(df$Date, df$GDP, main='GDP Moving Average', ylab='GDP', xlab='Year', type='l')lines(df$Date, df$MA10, col='blue')lines(df$Date, df$MA50, col='red')lines(df$Date, df$MA100, col='green')legend("bottomright", # Position of the legendlegend =c("GDP", "MA10", "MA50", "MA100"), # The text in the legendcol =c("black", "blue", "red", "green"), # The colorslty =1, # The line typescex =0.8) # The size of the legend text```## US Dollar IndexThe chart shows the US Dollar Index with its moving averages (MA10, MA50, MA100), reflecting a volatile currency market. Short-term fluctuations are closely followed by MA10, while MA50 and MA100 smooth out these fluctuations to reveal more sustained trends. A notable peak and subsequent decline around 2020 are followed by a recovery, with recent trends suggesting an increase in the dollar's value.```{r}df <-read_excel('data/data/dollar-index.xlsx')df <-na.omit(df)df$Date <-as.Date(df[[1]])df$USDIDX <-as.numeric(df[[2]])df$MA10 <-rollmean(df$USDIDX, 10, fill =NA, align ='right')df$MA50 <-rollmean(df$USDIDX, 50, fill =NA, align ='right')df$MA100 <-rollmean(df$USDIDX, 100, fill =NA, align ='right')plot(df$Date, df$USDIDX, main='US Dollar Index Moving Average', ylab='US Dollar Index', xlab='Year', type='l')lines(df$Date, df$MA10, col='blue')lines(df$Date, df$MA50, col='red')lines(df$Date, df$MA100, col='green')legend("bottomright", # Position of the legendlegend =c("USDIDX", "MA10", "MA50", "MA100"), # The text in the legendcol =c("black", "blue", "red", "green"), # The colorslty =1, # The line typescex =0.8) # The size of the legend text```## 30-Year Mortgage RateThe graph shows the trend of 30-year mortgage rates from around 1980 to 2020, with a clear peak in the early 1980s. It includes the actual rates and three moving averages (50-day, 100-day, and 500-day), which smooth out short-term fluctuations. Over time, there's a notable decline in mortgage rates, with all the lines converging, indicating stabilization of rates towards 2020.```{r}df <-read_excel('data/data/30-yrs-mortgage-rate.xlsx')df <-na.omit(df)df$Date <-as.Date(df[[1]])df$mort <-as.numeric(df[[2]])df$MA50 <-rollmean(df$mort, 50, fill =NA, align ='right')df$MA100 <-rollmean(df$mort, 100, fill =NA, align ='right')df$MA500 <-rollmean(df$mort, 500, fill =NA, align ='right')plot(df$Date, df$mort, main='30 Year Mortgage Rate Moving Average', ylab='MORT', xlab='Year', type='l')lines(df$Date, df$MA50, col='blue')lines(df$Date, df$MA100, col='red')lines(df$Date, df$MA500, col='green')legend("bottomright", # Position of the legendlegend =c("MORT", "MA50", "MA100", "MA500"), # The text in the legendcol =c("black", "blue", "red", "green"), # The colorslty =1, # The line typescex =0.8) # The size of the legend text```## Unemployment RateThe graph displays the historical trends of the unemployment rate from the 1960s to around 2020, including actual rates and their moving averages over 10, 50, and 100 periods. The chart shows several spikes indicating periods of increased unemployment, with the most significant peaks likely corresponding to economic recessions. Notably, there is an extreme spike in unemployment at the end of the timeline, suggesting a sudden, sharp increase in unemployment, possibly caused by the Coronavirus pandamic.```{r}df <-read_excel('data/data/unemploy.xlsx')df <-na.omit(df)df$Date <-as.Date(df[[1]])df$UER <-as.numeric(df[[2]])df$MA10 <-rollmean(df$UER, 10, fill =NA, align ='right')df$MA50 <-rollmean(df$UER, 50, fill =NA, align ='right')df$MA100 <-rollmean(df$UER, 100, fill =NA, align ='right')plot(df$Date, df$UER, main='Unemployment Rate Moving Average', ylab='Unemployment Rate', xlab='Year', type='l')lines(df$Date, df$MA10, col='blue')lines(df$Date, df$MA50, col='red')lines(df$Date, df$MA100, col='green')legend("bottomright", # Position of the legendlegend =c("UER", "MA10", "MA50", "MA100"), # The text in the legendcol =c("black", "blue", "red", "green"), # The colorslty =1, # The line typescex =0.8) # The size of the legend text```:::
